Bethe ansatz and Isomonodromic deformations

TL;DR

This paper explores the deep connections between Bethe ansatz solutions for the Gaudin model, geometric Langlands symmetries, and isomonodromic deformations, revealing new insights into integrable systems and special functions.

Contribution

It establishes a novel link between Bethe equations, Hecke operators, and Schlesinger transformations within the context of integrable models and geometric representation theory.

Findings

Identifies symmetries of Bethe equations with geometric Langlands operators.

Connects isomonodromic deformations to solutions of the Gaudin model.

Provides a unified framework for understanding Painlevé transcendents.

Abstract

We study symmetries of the Bethe equations for the Gaudin model appeared naturally in the framework of the geometric Langlands correspondence under the name of Hecke operators and under the name of Schlesinger transformations in the theory of isomonodromic deformations, and particularly in the theory of Painlev\'e transcendents.